1-Don't know what drawers you are talking about, but height isn't overly important. Just keep in mind that you want enough height to add cardboard furniture and have several inches left over above that. Personally, I use 45 or 50 gallon rubbermaid storage containers, but you can use something much smaller depending on how many roaches you want to produce.
2- If you figure that out for any insect, let me know, I'll be interested. Even when insects are contained in a feeding bowl somehow, I end up with occasional lizards climbing into the bowl and providing a ladder out for the bugs. For chameleons, I like to use small lightweight plastic food storage containers (deli cup sort of things) because I can just wedge these into the branches of their plants below the lizards where they can see the insects inside. These roaches cannot climb out of smooth sided containers like this. I've also used glass "dipping" bowls, glass salad bowls, and smooth crockery pet bowls (dollar store!).
3- They can eat anything crickets can eat, and really anything you want to put in there they will eat, including leftovers from your table as well. Some fresh veggies or fruit will keep them hydrated. I never use "water crystals" or similar- only fruit and veg and mine do great.
I use cardboard tubes from toilet paper, paper towels and giftwrap and cardboard egg flats and boxes for furniture.
4- In your house, you probably want some kind of cover. I don't use covers in my lizard building, but I do find especially adult male escapees. I didn't realize how many males got out until one year I had shelves made only of 2x4's with tubs on them. The top shelf was roaches, the next shelf down was empty waiting for baby lizards. I found perhaps a dozen or two male winged dubia roaches that had dropped from the tubs in the top shelf into the empty tubs below over the course of a month or so. They say males can't fly- I think they can somehow in the right conditions fly a little bit, or jump high over the top of a tub and flutter down or something. In those days I didn't use heat on my shelving though. I've noticed that now that I have heated shelves, I don't find many escapees ( a few though).
I don't know if I would recommend "drawers" if they pull in and out of a bank. You will want good ventilation for your roaches or they will smell and things can get moldy.
If you only want a small colony or two of roaches, you might want to get the tubs that some walmarts sell that have a seal and clamp lid. Cut the center of the lid out and glue (hot glue gun or silicone) screen over the center. Then you have a well ventilated, escape proof box. I did similar with my green banana roaches, although I limited ventiliation for those more. (By the way, recently I did a presentation on lizard farming to an FFA group at a high school. I discovered those green banana roaches are really great flyers and climbers when I opened my little bug box to feed some lizards and they climbed right up and out and took off flying 30 or 40' across the auditorium at some thrilled students!) height would not be as important if space is an issue, because the lids seal tight and roaches cannot climb out of these containers.
5- I would use a heat pad yes. I previously kept my roaches without heat and they slowed down in production during the winter. A heat pad set at 90 will keep them breeding and keep them in the tub where it is warm. If your room is cool year round, you won't get much production from your dubia without the heat pad. Contrary to some repeated beliefs, they will survive winter at room temp and even below- before I added the heated insect shelving, my lizard building used to drop down near 50 at nights and climb only into the 60s for a month or two during mid winter. My dubia roaches survived just fine. But I don't think they did much breeding during that time.
